What parameters can be tweaked to avoid a generator or discriminator loss collapsing to zero when training a DC-GAN?
GANs are notably hard to train and it is not uncommon to have large bumps in the losses. The learning rate is a good start but the instability may come from a wide variety of reasons. I'm assuming that you have no bug in your code or data.

For one, gradient descent is not well suited to the 2-player game we're playing. I've personally found ExtraAdam to yield much more stable training (code, paper). 

It could also come from the loss and many tricks have been developed and one of the most popular one is enforcing smoothness in the gradient (see W-GAN, W-GAN-GP etc.). SpectralNorm (code paper) is a very popular and recent normalization technique for the discriminator.

There are a number of additional tricks to make GANs work like label smoothing and flipping, different update rates for the discriminator and generators (as in BigGAN for instance). I suggest you have a look at this nice repo of (somewhat seasoned) tricks: ganhacks.